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Collective action problem in
heterogeneous groups
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Sergey Gavrilets1,2,3
1
Department of Ecology and Evolutionary Biology, 2Department of Mathematics, and 3National Institute for
Mathematical and Biological Synthesis, University of Tennessee, Knoxville, TN 37996, USA
Review
Cite this article: Gavrilets S. 2015 Collective
action problem in heterogeneous groups. Phil.
Trans. R. Soc. B 370: 20150016.
http://dx.doi.org/10.1098/rstb.2015.0016
Accepted: 21 August 2015
One contribution of 13 to a theme issue
‘Solving the puzzle of collective action through
inter-individual differences: evidence from
primates and humans’.
Subject Areas:
behaviour, evolution, genetics,
theoretical biology
Keywords:
altruism, cooperation, mathematical modelling,
collaboration, evolutionarily stable strategies
Author for correspondence:
Sergey Gavrilets
e-mail: [email protected]
I review the theoretical and experimental literature on the collective action problem in groups whose members differ in various characteristics affecting
individual costs, benefits and preferences in collective actions. I focus on evolutionary models that predict how individual efforts and fitnesses, group
efforts and the amount of produced collective goods depend on the group’s
size and heterogeneity, as well as on the benefit and cost functions and parameters. I consider collective actions that aim to overcome the challenges
from nature or win competition with neighbouring groups of co-specifics.
I show that the largest contributors towards production of collective goods
will typically be group members with the highest stake in it or for whom
the effort is least costly, or those who have the largest capability or initial
endowment. Under some conditions, such group members end up with smaller net pay-offs than the rest of the group. That is, they effectively behave as
altruists. With weak nonlinearity in benefit and cost functions, the group
effort typically decreases with group size and increases with within-group
heterogeneity. With strong nonlinearity in benefit and cost functions, these
patterns are reversed. I discuss the implications of theoretical results for
animal behaviour, human origins and psychology.
1. Background
Group living, which is widespread in animals, has both fitness benefits and costs,
thus group-living species have developed various adaptations for dealing with
their social environments [1–3]. Group-living animals necessarily have to coordinate many of their activities such as eating, sleeping or moving. In some species,
individuals also actively collaborate with their group-mates, for example in territorial defence, hunting or breeding. In particular, our ancestors were involved in
collective large-game hunting and between-group conflicts over territories,
mating and other resources. Because of the economy of scale, within-group collaboration can be very profitable, especially for humans. In modern humans,
collaboration happens at all levels of human society.
Group-living implies interdependence, collective actions, shared benefits and
shared costs. Interdependence and shared interests, however, do not eliminate
conflicts of interests of group-mates. Even when cooperation benefits all partners,
they will probably end up not cooperating because they can see the advantages of
free-riding or fear the dangers of being exploited by others who may choose to
free-ride. As a consequence, defection becomes a dominant strategy in the Prisoner’s Dilemma [4], individuals withdraw their efforts in production of a collective
good [5], or members of a commune fail to exercise self-restraint in exploitation
of a communal resource, destroying it as a result [6]. Adam Smith’s powerful
metaphor of the ‘invisible hand’ emphasizes unintended social benefits that can
result from individuals’ pursuit of personal interests. However, in the words of
Russell Hardin, when engaged in collective actions ‘. . .all too often we are less
helped by the benevolent invisible hand than we are injured by the malevolent
back of that hand; that is, in seeking private interests, we fail to secure greater
collective interests’ [7, p. 6].
How various social dilemmas, i.e. conflicts between individual and collective
interests, can be resolved has been a subject of intense observational, experimental
and theoretical work across a variety of scientific disciplines including economics,
& 2015 The Author(s) Published by the Royal Society. All rights reserved.
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Olson’s Logic has had a tremendous impact on a diversity of
social sciences (see [7–12] for summaries of more recent
work). However, it appears to be somewhat ignored in evolutionary biology. (According to the Web of Science database,
the book has been cited more than 7500 times. Of these, less
than 1% is in the evolutionary biology literature.) Nevertheless,
the ideas from economics have percolated into theoretical evolutionary biology and have been developed both further and
independently into an impressive array of new directions. By
now, it is well appreciated that increasing the group size typically makes cooperation more difficult (Olson’s first main
insight) and that punishment is an efficient way to promote
cooperation (a part of Olson’s third main insight). However,
the effects of within-group heterogeneity (Olson’s second
main insight) are much less understood.
(a) Review goals
My general goal here is to review the theoretical work on
the CAP that might be particularly interesting and relevant
for behavioural evolutionary biologists, anthropologists and
psychologists. The important role of the CAP in undermining
within-group cooperation in between-group territorial conflicts
is well recognized by evolutionary primatologists [13–19]
as well as by some other behavioural biologists [20–23].
In the anthropological literature, the existence of the CAP
in between-group conflicts [24–26] as well as in hunting and
other types of collective production [27–29] is also well
acknowledged. In particular, the facts that foragers apparently
have ‘limited needs’ and that their work efforts are surprisingly modest have been linked to their culture of sharing, which
makes food and other objects public goods [27,30]. Earlier
discussions by Hawkes [27] and Nunn & Lewis [15] used
simple two-player games to illustrate the CAP. While there is
indeed a substantial theoretical literature on dyadic interactions
[31], two-player games are not necessarily very informative
about multiplayer dynamics. I will be looking at collective
(b) Alternative approaches
There are a number of alternative theoretical approaches for
studying collective actions which will not be covered in this
study. I briefly discuss them here. I will be concentrating on
2
Phil. Trans. R. Soc. B 370: 20150016
— the group-size paradox: collective goods provisioning
decreases with the group size;
— exploitation of the great by the small: the largest beneficiary of the collective good bears a disproportionately
large burden of its production; and
— selective incentives (i.e. reward and/or punishment) and
institutional designs can help large groups overcome
the CAP.
actions involving more than two individuals simultaneously
in further sections.
An inherent feature of most animal and human groups
involved in collective activities is their heterogeneity. Group
members may differ in how much they value the collective
good, how many resources they have, how much cost they pay
per unit of effort and so on. Olson [5] argued that this heterogeneity would often lead to the situations where ‘there is a systematic
tendency for ‘exploitation’ of the great by the small’ (p. 29). That
is, some individuals (e.g. those with the highest stake in the
collective good or for whom the effort would be least costly, or
those who have the largest initial endowment) would contribute
all the effort while the rest of the group-mates will free-ride
contributing nothing. (In Olson’s terminology, a group with
individuals whose benefits from a collective action exceed the
associated costs even if they are solely borne by these individuals
is a ‘privileged group’.) Subsequent theoretical work, mostly in
the economics literature, explored this issue in detail (see
below). However, as noted earlier, the insights from this work
have not been applied much to biological and anthropological
problems (although the potential importance of within-group
heterogeneity in biological CAPs was recognized by Nunn &
Lewis [13,15]). Below I will discuss the relevant theoretical and
experimental results in detail.
Even though the amount of work on the CAP published in
the economics literature is enormous, there are certain limitations on its applicability to biological questions. While
the analysis of economics models typically focuses on Nash
equilibria (NE, at which individuals aim to maximize their
absolute gain), biological models usually study evolutionarily
stable strategies (ESS, at which natural selection maximizes
the individuals’ relative gain). NE and ESS can lead to rather
different predictions [32–34]. Economics models often ignore
the question how individuals can identify an appropriate NE,
whereas in evolutionary models the question of convergence
to an ESS is of primary importance. Moreover, many NE
identified by game theory methods are evolutionarily irrelevant [35, ch. 5]. Economics models also typically ignore
many evolutionary factors, such as group selection, genetic
relatedness, spatial structure and migration, which all can significantly affect evolved social behaviour. These considerations
imply that additional studies of biologically inspired models
are often needed.
My main focus will be on models relevant for evolutionary
biology within a broad cross-specific perspective. I will pay
special attention to the evolution of social instincts, that is,
genetically based propensities that govern the behaviour of
individuals in social interactions. As first argued by Darwin
[36], social instincts have evolved by natural and sexual
selection. In modern perspective, these instinctive social behaviours are plastic, resulting from the interaction of the genotype
with the social environment [37]. Within this framework, individual behavioural strategies change by random mutation (and
recombination). However, some of the conclusions emerging
from using this approach may also be valid for other strategy
revision protocols used in evolutionary game theory, such as
learning or myopic optimization [38].
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evolutionary biology, anthropology and psychology. Here,
I will be concerned with one particular set of social
dilemmas—the collective action problem (CAP) within the
context of collective goods production [5,7]. This is a situation when group members can make an individually costly
effort towards achieving some group-beneficial goals
(e.g. hunting a large game or defending the territory) but
because individuals potentially can receive the benefits without having contributed to their production, they have an
incentive to reduce their effort or withdraw it completely.
If enough individuals fail to contribute, the collective benefit
is not produced and everybody suffers.
Systematic theoretical studies of collective action started
50 years ago with a publication of Mancur Olson’s book
The Logic of Collective Action in 1965 [5]. Olson’s book offered
three main insights/hypotheses:
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2. Mathematical models and their predictions
Let us consider a population of individuals subdivided into G
groups each of size n individuals. Assume that individual i
in group j makes an effort xij towards the group’s success
in a collective action. Effort xij can be treated as a binary variable (e.g. taking only two values: 0 and 1) or a non-negative
continuous variable. Individual efforts of group members are
aggregated into a group effort Xj. Following the economics literature, I will call the function converting individual efforts
x1j , . . . , xnj into group effort Xj the impact function. The most
common impact function in the literature is linear:
X
Xj ¼
xij :
ð2:1aÞ
i
Two other examples are the ‘weakest-link’ when Xj is equal to
the minimum of xij values in the group and the ‘best-shot’
when Xj is equal to the maximum of xij values. In some
models, individuals differ in their strengths and/or capabiliP
ties si and the impact function is defined as Xj ¼ i si xij , so
that individuals with largest si values have the largest effect
on Xj [77,78]. Some work uses the ‘constant elasticity of
substitution’ (CES) impact function [79,80]
!a
X 1=a
Xj ¼
xij
,
ð2:1bÞ
i
(c) Review structure
The literature on multiplayer evolutionary games and
collective action is quite diverse [67,68]. In the case of dyadic
interactions with just two discrete strategies, all possible
games can be classified in an unambiguous way. With
continuous strategies and/or with multiple players such a
classification becomes much more difficult [69,70]. Rather
than trying to use such a classification, I will structure my
discussion around two general types of social interactions
which I will call ‘us versus nature’ games and ‘us versus
them’ games [71]. These games correspond to two general
types of collective actions in which our ancestors were almost
certainly engaged. The first type includes group activities
such as defence from predators, some types of hunting or
food collection, use of fire, etc. The success of a particular group
in these activities largely does not depend on the actions of
neighbouring groups. The second type includes direct conflicts
and/or competition with other groups over territory and other
resources including mating. The success of a particular group
which will play an important role in discussions below.
Here a is a non-negative parameter which can be interpreted
as a measure of within-group synergicity or collaborative
ability [71]. If collaborative ability a is very small (a1),
the group is only as efficient as its member with the largest
effort (Xj maxi ðxij Þ). Increasing collaborative ability a
while keeping individual efforts the same increases the
group effort Xj. If all group members make an equal effort x,
then Xj ¼ na x: The latter function is related to the Lanchester–
Osipov model used in military research [81,82] and also in
evolutionary biology [83,84].
I will specify the individual pay-off as
fij ¼ fij,0 þ bij Pj cij xij :
ð2:2Þ
Here, fij,0 is a baseline pay-off, bij the value of group j’s success to individual i and cij the cost parameter for this
individual. Function Pj is a non-decreasing function of Xj
and, potentially, of other groups’ efforts; it gives the normalized value of the resource produced by group j as a result of
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Phil. Trans. R. Soc. B 370: 20150016
in an ‘us versus them’ game definitely depends on the actions
of other neighbouring groups. The outcomes of both types of
collective actions strongly affect individual reproduction as
well as group survival. ‘Us versus nature’ games comprise
typical collective action models with linear or nonlinear payoff functions [72–75]. ‘Us versus them’ games correspond to
models of between-group contests in economics theory [76].
Below I start by outlining a general approach for modelling
collective actions in evolutionary biology. Then I consider ‘us
versus nature’ and ‘us versus them’ games separately. For
each type of games, I summarize results on homogeneous
groups before discussing the behaviour of models describing
heterogeneous groups. I will illustrate analytical results
using a series of numerical examples based on a simple
model linking the two types of games. I will also review
relevant experimental economics research.
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models describing the CAP in situations when group members pay individual costs to share group benefits. For
discussions of models describing the ‘tragedy of the commons’ [6], when group members receive individual benefits
but share costs, see e.g. [39– 43]; some relevant empirical
evidence on the effects of heterogeneity is reviewed in
[44,45]. I will consider models focusing on NE and ESS; for
related work using other equilibrium concepts, e.g. those
that include errors in decision-making or learning, see
[38,46 –50]. I will focus on models explicitly allowing for a
single type of individual effort such as voluntarily contributing
to production of a collective good. More complex models that
allow for multiple types of individual efforts simultaneously
(e.g. contributing to production and punishing defectors
[51–54]) are outside of the scope of my review. I will assume
that groups are formed exogenously. For recent reviews on
endogenous coalition formation in the theoretical and biological literatures, see [55] and [56], respectively. In the models to
be discussed, exact spatial arrangement of groups is largely
irrelevant. Public goods games (PGG) in spatially structured
populations have been recently reviewed in Perc et al. [57].
In models to be discussed below, individuals will make
their decisions about the actions to take simultaneously.
There is, however, an alternative approach to modelling
collective actions which explicitly assumes that individuals
make decisions about their efforts sequentially [9,58–64].
In this case, the group behaviour may exhibit a threshold
effect: once there is a critical mass of contributors, everybody
else also starts contributing. The ‘first movers’ can be individuals with the highest interest in the collective good. In some
of these models, group members are basing their decisions not
(only) on the benefit/cost considerations but rather on a simple
behavioural rule: contribute and/or participate if a certain
number or proportion of other individuals are already participating (‘bandwagon effect’) [58]. These models have been used
for describing social mobilization, formation and diffusion of
social movements, spread of social unrest and contagion
through social networks while allowing for stochasticity in
decision-making, effects of learning beneficial strategies and
a possibility that some individuals (e.g. ‘organizers’) promote
cooperative behaviour of others [59,61–63,65,66]. This work
is outside of the scope of my review.
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ð2:3aÞ
where f is the average pay-off in the whole population. Assuming that group j contributes a share Sj of offspring to the next
generation and that individuals compete for reproductive success within the groups, wij ¼ Sj ð fij =fj Þ, where fj is the average
pay-off in group j. With high-risk collective actions and
between-group competition, group extinction becomes possible. If the probability that group j survives between-group
competition and contributes to the next generation is proportional to the degree of success Pj in a collective action and
the spots of groups going extinct are repopulated by duplicating
surviving groups, then
Pj fij
wij ¼ ,
P fj
ð2:3bÞ
is the average of Pj.
where P
These models describe multi-level selection operating
at both within- and between-group levels [86]. Group-level
selection favours large individual efforts xij (which would
increase the probability/degree of group success Pj ), while
individual-level selection favours low efforts xij (which
would reduce the individual costs term cxij ) creating an
incentive to free-ride. What makes it beneficial to free-ride
is that all group members share the benefit of high Pj
values at least partially independently of their individual
contributions to the group’s success. Using fitness function
(2.3a) implies that groups are formed randomly each generation. In the case of animal and human groups, this
assumption is difficult to justify biologically, but it leads to
simpler mathematics and has been used in the majority of
theoretical papers on cooperation in evolutionary biology.
Fitness function (2.3b) is more appropriate if groups retain
their identity from generation to generation as is the case in
most biological populations.
(a) Us versus nature games
In us versus nature games, the probability/degree Pj of group
j success depends on the group effort Xj but does not depend
on the efforts of other groups. Most published modelling
work corresponds to fitness function (2.3a) and uses an
assumption (implicitly or explicitly) that groups are formed
randomly at the beginning of each generation.
First, I discuss the evolutionary dynamics in the case when
groups are homogeneous (see [31,68] for recent thorough
reviews). In such groups, we can set fij,0 ¼ 1, bij ¼ b and
cij ¼ c. In the simplest case of linear impact and production
functions, also known as an N-player Prisoner’s Dilemma
game, fi ¼ 1 þ ðk=nÞX cxi , where k is a reward factor (by
which the group contribution X is multiplied before the
reward is divided between n group members). (Here and
below I drop subscript j specifying the group both for simplicity of notation and also because economics models usually
focus on a single group while evolutionary biology models
usually assume that groups have identical structure). In this
model, group members would jointly benefit if they all
make a positive effort x as long as k . c. However, the logic
of the Prisoner’s Dilemma prevents this from happening [31]
and the groups end up contributing nothing (unless the
reward is very high, k . nc, so that the pay-off from one’s
own action, k/n exceeds the cost c).
Although the N-player Prisoner’s Dilemma game is the
most popular model for studying cooperation in groups, it is
by no means the only reasonable model [68]. In the Volunteer’s
Dilemma model [72], it is assumed that a single volunteer can
produce a collective good of value b to everybody (which corresponds to a step production function). Naturally, each group
member prefers others to pay the cost of production. In this
model, if b . c, there is a symmetric mixed equilibrium with
the probability of defection being ðc=bÞ1=ðn1Þ , 1: Weesie &
Franzen [87] showed that cost sharing among contributors
increases the probability of volunteering. Archetti [74] has
generalized this model in a number of directions including
cases when individual benefit is an S-shaped function of the
total number of volunteers and when group members are
genetically related. The latter factor increases the probability
of volunteering.
The Volunteer’s Dilemma is related to the Snowdrift
game as well as to the Hawk–Dove and Chicken games
[31], in all of which cooperation is a preferable strategy if
the opponent does not cooperate. Zheng et al. [88] studied
an N-person Snowdrift game with cost sharing among contributors. They showed that cooperators can be maintained in
the population at a small frequency (approximately 1/n
with large n). Similar conclusions emerge in a version of an
N-person Stag Hunt game in which it pays to contribute to
a public good if there is a certain number of other contributors [89]. For generalizations to other types of production
functions, finite population size and variable population size,
see e.g. [90–93]. All of the above work assumed binary individual efforts, i.e. xij could be only 0 or 1. If individuals’
efforts are continuous and costs and/or benefits are certain
nonlinear functions of the efforts, then at a NE group
members are predicted to make a certain non-zero effort
[94 –96]. Doebeli et al. [97] studied a continuous multiplayer Snowdrift game with quadratic production and cost
functions. They showed that under some conditions, the
population evolves either to a monomorphic state where
everybody contributes a small effort or it evolves, via an evolutionary branching [98], to a dimorphic state where some
individuals contribute a large effort while the rest free-ride.
Frank [85] studied a model with multiplicative interactions
of individual costs and group benefits. In his model, it is
beneficial to make an effort if nobody else in the group
4
Phil. Trans. R. Soc. B 370: 20150016
fij
wij ¼ ,
f
(i) Homogeneous groups
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collective effort; we normalize Pj relative to a maximum possible reward size (0 Pj 1) so that it measures the success of
collective action (actual or expected). Using economics terminology, I will call this function the production function in us
versus nature games and contest success function in us
versus them games. In some models, one assumes that individual i in group j gets share vij of the total prize Bj, so that
bij ¼ vij Bj : If group members share the reward equally,
vij ¼ 1=n: In equation (2.2), the benefit and cost terms contribute additively to individual pay-off. More complex functions
can also be used (for example, multiplicative [39,85]). Note
also that equation (2.2) assumes that individual cost, cijxij, is
linear in individual effort xij but nonlinear functions are
also used often (see below).
Individual pay-offs define individual fitness (i.e. reproductive success) wij. There are different ways to specify wij. For
example, when individuals compete over the whole population
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Within-group heterogeneity can be introduced in a number
of different ways. Earlier work (reviewed in [8]) focused
on heterogeneity in initial endowment/wealth of group
members and used linear impact, production and cost
functions. This corresponds to pay-off being written as
fi ¼ fi,0 þ ðk=nÞX cxi , where fi,0 is the baseline pay-off
(endowment) of individual i. An earlier theoretical study
[99] predicted an increase in total public good production
as the distribution of endowments becomes more unequal.
Andreoni [100] showed that at a NE, the contributors will
include only the wealthiest individuals (i.e. individuals
with fi,0 larger than a certain threshold). In the limit of a
very large group size, there will be just a single contributor—the wealthiest group member. These results, which
supports Olson’s logic, are conditioned on the use of
linear functions. In Frank’s model [85] with multiplicative
interactions of costs and benefits, individuals contribute
proportionally to their endowments, whereas the group
effort does not depend on the distribution of endowments. Moreover, in his model all individuals have an
equal fitness/pay-off at equilibrium, independently of their
initial endowment.
A substantial effort has focused on models with heterogeneity in valuation bi and cost ci parameters. With some
nonlinear production and/or cost functions, all group members are predicted to make a contribution and the group
effort does not necessarily increase with within-group
inequality/heterogeneity; see [79] and [101] for examples
with n ¼ 2. McGinty & Milam [102] analysed a case with
an arbitrary group size assuming quadratic benefit and cost
functions. They predicted that individuals will contribute in
proportion to their ratios, bi/ci, of valuation to cost parameters. Some recent work analysed an asymmetric version
of the Volunteer’s Dilemma assuming that individuals
differ with respect to how much they value the public good
(bi) and how much cost (ci) each would pay to produce it.
In particular, Diekmann [73] identified a mixed strategy equilibrium at which the probability of defection is proportional
to bi/ci, so that ‘strong’ players (i.e. with higher valuation
and/or lower cost) will be more likely to defect (see [103]).
This solution is counterintuitive in that it directly contradicts
Olson’s conjecture about the exploitation of the great by the
small. He et al. [104] generalized results in [73] for the case
of genetically related individuals in a simplified version in
which there is one ‘strong’ player, with benefit and cost
(iii) Examples
I will attempt to illustrate and clarify the above diversity of
theoretical results using a simple model as a reference
point. Besides its simplicity, this model also allows for comparisons between us versus nature games and us versus
them games to be discussed below. Specifically, I define the
degree of success (production function) for group j by a
saturating function
Pj ¼
Xj
,
X j þ X0
ð2:4Þ
[71]. Here, X0 is a parameter specifying the group effort Xj at
which the degree of success is 50%; the larger X0, the more
group effort Xj is needed to secure the success. I will call X0
the half-success effort. I will assume that within each group
individuals have equal baseline pay-offs, fi,0 ¼ 1, but differ in
valuation bi and cost ci parameters and that groups have
identical distributions of these parameters among their
members. Below I will consider how the predicted individual
efforts, pay-offs and fertilities, group efforts and degree of
success in producing the collective good depend on the
group size, within-group heterogeneity, half-success efforts,
the shape of impact and cost functions, and benefit and cost
parameters. I will present analytical results and will also
illustrate them numerically.
In deriving analytical approximations, I used an invasion
analysis/adaptive dynamics approximation assuming that
within-population genetic variation is very low ([98,106]; see
[71,77] for applications of these methods to similar models).
Numerical simulations show that theoretical conclusions
remain valid at a qualitative level even in the presence of
some genetic variation. The derivations are straightforward,
so to save space I do not show them here. The equations I do
show assume, for simplicity, that the number of groups G is
very large.
In analytical approximations, models’ behaviour depends
on certain individual-specific benefit-to-cost ratios, which
I will denote as ri. To describe the effects of within-group
heterogeneity in ri, it is convenient to use a heterogeneity
measure based on the notion of the generalized mean.
5
Phil. Trans. R. Soc. B 370: 20150016
(ii) Heterogeneous groups
parameters bs and cs, and n21 identical ‘weak’ players, with
parameters bw and cw, where bs =cs . bw =cw : They showed
that genetic relatedness between group-mates increases
cooperation. In a follow-up paper, however, He et al. [105]
argued that the mixed equilibrium identified by Diekmann
[73] is evolutionary unstable. They also showed the existence
of two other ESSs. At one ESS, the collective good is produced
by the strong player while all n21 weak players defect. At the
other ESS, the strong player always defects while weak players
defect with a certain probability. He et al. [105] argued that
the former equilibrium has a larger domain of attraction and
therefore is more relevant biologically.
A general, and intuitive, conclusion of this body of work
is that individual efforts increase with individual benefit/cost
ratios and endowments. Linear models exhibit a threshold
effect so that group members contribute to production only
if they are above a certain minimum value in a relevant
characteristic. In nonlinear models, all individuals contribute
but to a different extent, with some contributions being
very low. Group efforts usually increase with within-group
inequality but there are exceptions.
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contributes. Frank showed that individuals always make a
positive contribution which is inversely proportional to the
group size. Allowing for within-group genetic relatedness
or an accelerated increase in costs and benefits with efforts
increases individual efforts [85].
A general prediction of nonlinear models of homogeneous
groups is thus that under some conditions groups will produce
collective goods. With discrete strategies, there will be a
small proportion of contributors while the rest will free-ride.
With continuous strategies, the groups will be represented
either by mild cooperators (i.e. individuals making small
contributions) or by a mixture of a small number of cooperators making substantial contributions and a large number
of defectors. Increasing group size typically decreases
within-group cooperation.
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0.5
share of benefit vi
0.4
0.3
0.2
0.1
1
2
3
4
5
6
7
8
rank i
Figure 1. Dependence of the share of benefit vi going to an individual of
rank i onPparameter d in the models used in numerical simulations with
n ¼ 8.
vi ¼ 1: (Online version in colour.)
Given a set of individual characteristics r1,. . .,rn andqa
P 1=q
parameter q, the generalized mean is Mq ¼ ð1=nÞ i ri
:
In this notation, the arithmetic mean is M1, the harmonic
mean is M21 and so on. Below I will use a normalized version
of the generalized mean, Hq ¼ Mq =M1 : If all individuals are
identical (i.e. ri ¼ r), Hq ¼ 1 for all q. As one introduces variation in ri, Hq decreases if q . 1, but increases if q , 1.
Parameter Hq thus captures the effects of within-group heterogeneity. (It also depends on the group size n, but this
dependence is relatively weak).
In numerical simulations, following Gavrilets &
Fortunato [77], I posit that individuals have identical cost
parameters (ci ¼ 1) but differ in benefit parameters. I specify
the latter as bi ¼ vi B: I assume the total benefit per group B is
fixed and that individuals are ranked according to the share
vi of B they receive. This share is set to be proportional to a
power function of individual rank i: vi ðn þ 1 iÞd : Here
d is a parameter measuring the degree of inequality: with
d ¼ 0, all shares are equal to 1/n and the groups are egalitarian; with d ¼ 1, share vi is a linear function of individual rank
i; and with d . 1, it is a decreasing, concave-up function of
rank. In simulations, I used five values of d ¼ 0.25, 0.5, 1, 2
and 4. The ranks were assigned randomly at the beginning
of the generation (e.g. based on strengths). Figure 1 illustrates
the effect of d on shares vi. For simplicity of interpretation, in
my numerical results within-group heterogeneity will be captured by d rather than by Hq. In implementing the model
numerically, I assumed, following earlier work [84,77,107],
that individual efforts at each rank were controlled by different loci, so that in each individual only one locus was
expressed, corresponding to his rank. Generations were discrete and non-overlapping. I allowed for mutation, free
recombination and migration. There were two sexes but
only one of them (males) was engaged in collective actions.
Group selection was captured by making each group in the
new generation to independently descend from a group in
the previous generation with probability proportional to Pj.
In surviving groups, each female produced two offspring
while reproductive success of males was proportional to
their pay-off fij. In models with no group extinction, all offspring dispersed randomly and groups were formed
randomly at the beginning of each generation. In models
with group extinctions, male offspring stayed in their native
6
Phil. Trans. R. Soc. B 370: 20150016
0
groups while females dispersed randomly (as in chimpanzees
and probably our ancestors [108]). See Gavrilets & Fortunato
[77] for additional details of simulation methods.
Basic model. First, consider the case with no group extinction
(i.e. using fitness function (2.3a)) and linear cost and impact
functions. It is useful to define parameters ri ¼ bi =ðci X0 Þ
equal to the individual benefit divided by the cost at halfsuccess effort, and use a normalized group effort Z ¼ X=X0 :
Let individuals be ranked according to ri and there be no ties.
In this model, the population evolves to a state at which
within each group only an individual with the highest benefit
to cost ratio, r1, can make an effort while all other group-mates
always contribute nothing. Rank-1 individual contributes only
pffiffiffiffi
if r1 . 1; his effort (and that of the group) then is Z ¼ r1 1:
Note that the condition r1 . 1 can be rewritten as a requirement for the share of the reward going to the rank-1
individual to be high enough: v1 . vcrit ¼ c1 X0 =B: In this
case, the highest valuator benefits from producing the
collective good even if acting alone.
Figure 2a illustrates these results for the case of no variation
in cost parameters (ci ¼ 1). The figure shows that the group
effort increases with increasing within-group inequality and
decreasing group size; both these factors increase the share of
the highest valuator v1. With n ¼ 8 and low inequality or
with n ¼ 16, contributions are practically absent except for
small ‘noise’ due to recurrent mutation. Allowing for group
extinction (i.e. using fitness function (2.3b)) changes the
dynamics considerably (figure 2b). Now, all individuals with
valuations above a certain threshold vcrit make non-zero contributions which increase linearly with their valuations vi. The
group effort X weakly depends on within-group heterogeneity
and group size. Note that in spite of within-group inequality in
shares of the reward, variation in fertilities is relatively modest,
both with and without group extinction. Note also that under
some conditions the highest valuator has lower fertility than
his ‘subordinate’ group-mates (e.g. with group extinction).
Nonlinear costs. Second, assume that the cost term in pay-off
equation (2.2) is cxgij with g . 1. This implies that costs of
small efforts are insignificant but costs rapidly increase as efforts
become larger. This model’s behaviour is qualitatively different
from that with linear costs. Now, the relevant benefit-to-cost
P
ratios are ri ¼ bi =ðci X02 Þ and R ¼ ri : (Note that with no variation in individual costs, i.e. if ci ¼ c, R is equal to the total
P
benefit B ¼ bi divided by the total cost at half-success rate
cX02 :) There is no minimum valuation anymore for making a
contribution and each individual makes a positive effort pro1=ðg1Þ
portional to ri
: The normalized group effort Z increases
monotonically with the product of the benefit-to-cost ratio R, a
power function of group size, ng2 , and heterogeneity parameter Hg1 computed using ri values. (Specifically, Z solves
the equation gZg1 ðZ þ 1Þ2 ¼ Rng2 Vg1 :) With quadratic costs
(i.e. if g ¼ 2), the group effort depends neither on the group
size nor on within-group heterogeneity, and individual contributions are linearly proportional to ri. With 1 , g , 2, the
group effort decreases with group size n and increases with variation in ri. With, g . 2, the situation is reversed: the group effort
increases with group size n and decreases with variation in ri.
Figure 3a,b illustrates these results for the case of no
variation in cost parameters (ci ¼ 1). Allowing for group extinction (i.e. using fitness function (2.3b)) increases individual
efforts (figure 3c,d). The effects of within-group inequality
greatly decrease, while those of the group size diminish for
g ¼ 1.5, but are augmented for g ¼ 2.5.
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(b)
1.5
1.0
0.5
0
n=4
n=8
n = 16
0.5
0
n=4
n=8
n = 16
n=4
n=8
n = 16
1.0
fertility
fertility
1.0
7
1.5
1.0
0.5
0
n=4
n=8
0.5
0
n = 16
(b)
1.0
efforts
efforts
(a)
0.5
0
n=4
n=8
0.5
0
n = 16
(d) 3
2
efforts
n=8
efforts
n=4
3
1
n=4
n=8
n=4
n=8
n = 16
n=4
n=8
n = 16
n=4
n=8
n = 16
1
1.0
fertility
fertility
n = 16
2
0
n = 16
1.0
0.5
0
n=8
0.5
(c)
0
n=4
1.0
fertility
fertility
0.5
0
n = 16
1.0
0
1.0
n=4
n=8
n = 16
0.5
0
Figure 3. Effects of nonlinear costs in the ‘us versus nature’ game. First column (a,c): g ¼ 1.5, second column (b,d): g ¼ 2.5. (a,b) No group extinction, (c,d ) with
group extinction. (Online version in colour.)
Synergicity. Third, instead of an additive impact function
let us use the CES function (2.1b) with collaborative ability a.
The behaviour of this model depends on the benefit-to-cost
P
parameters ri ¼ bi =ðci X0 Þ and R ¼ ri , a power function of
group size na2 and heterogeneity parameter Ha1 computed
using the corresponding ri values. If Rna2 Ha1 , 1, the
groups make no effort. Otherwise individual contributions
a=ða1Þ
are proportional
and the normalized group effort
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffito ri
a
2
Z ¼ Rn Ha1 1: In a special case of a ¼ 2, the group
effort does not depend on the group’s size or heterogeneity,
and individual contributions are proportional to r2i : With
1 , a , 2, the group effort decreases with group size n and
increases with variation in valuations vi. With, a . 2, the
situation is reversed: the group effort increases with group
size n and decreases with variation in valuations vi.
Figure 4a,b illustrates these results for the case of no variation
in cost parameters (ci ¼ 1). With group extinction (i.e. using
fitness function 2.3b), individual efforts greatly increase
(figure 4c,d). The effects of within-group inequality greatly
decrease, while those of group size diminish for a ¼ 1.5, but
are augmented for a ¼ 2.5. Note that with synergicity (i.e. with
a . 1), groups can make significant total effort with much smaller individual efforts, which explains why the height of bars in
figure 4 is smaller than that in figure 3.
Conclusions on examples. In the basic model, individuals
defect if rewards are small but cooperate in securing big
rewards. Increasing the reward size causes an increase in
the efforts of high valuators but it can also decrease the
efforts of low valuators who would increasingly free-ride.
Increasing inequality decreases the efforts of low valuators
but this is overcompensated by increased efforts of high
valuators. As a result, with linear cost and impact functions,
the group effort increases with inequality and decreases with
group size. With linear costs, there is a threshold effect:
Phil. Trans. R. Soc. B 370: 20150016
Figure 2. Collective action in the ‘us versus nature’ game with production function (2.4) without (a) and with (b) group extinction. Additive impact and cost
functions. Summary results over all runs for one set of parameters, with B ¼ 4, X0 ¼ 1, c ¼ 1, G ¼ 1000. For each run, the values are averages over individuals
of rank i in all groups in the population. Colours show the relevant amounts for individuals of different ranks, from the rank-1 individual at the bottom (red) to the
rank-n individual at the top. Each set of bars corresponds to a specific value of group size n. Each bar within a set corresponds to a specific value of within-group
inequality d, from the smallest on the left (d ¼ 0.25; low inequality) to the largest on the right (d ¼ 4; high inequality). Top half
P of graphs: individual efforts xi;
fj ). (Online version in colour.)
the total height of each bar gives the group effort. Bottom half of graphs: fertilities, i.e. individual shares of reproduction ( fi =
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(a)
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1
efforts
1.0
0.5
0
(b)
n=4
n=8
n = 16
1.0
0.1
0
fertility
n=8
0
10
2
1
1
n=4
n=8
n = 16
(d)
n=4
n=8
n = 16
2
0.1
10
1
0
0.1
1
n=4
n=8
n = 16
n=4
n=8
n = 16
0.1
1.0
fertility
1.0
fertility
n = 16
0.5
n = 16
efforts
fertility
efforts
n=4
3
0.5
0
n=8
1.0
0.5
0
n=4
n=4
n=8
n = 16
0.5
0
Figure 4. Effects of synergicity in the ‘us versus nature’ game. First column (a,c): a ¼ 1.5, second column (b,d): a ¼ 2.5. (a,b) No group extinction, (c,d) with
group extinction. The dashes show the group effort on the logarithmic scale specified on the right y-axis. (Online version in colour.)
individuals contribute only if their valuation is above a certain critical value. In nonlinear models, all individuals
contribute to public goods production. With linear costs,
there is high dispersion of efforts within groups. However,
this dispersion is reduced with quadratic costs and, especially,
with synergicity. With highly nonlinear cost and benefit functions, the group effort can increase with group size and
can decrease with inequality. Allowing for group extinction
results in two main effects. First, all group members contribute
proportionally to their valuations. Second, individual and
group efforts as well as the degree/probabilities of success
significantly increase. In most cases, individual share of reproduction grows with rank/valuation. But there are some cases
where highest valuators (who are simultaneously the biggest
contributors) have lower fertility than other individuals
because of the costs paid.
(iv) Experimental games
Us versus nature games with homogeneous groups have
been studied intensively in experiments. In linear PGG,
contributions typically show a continuous decline [109]. However, in nonlinear PGG, contributions are often maintained at
intermediate values, as predicted by models [96].
There has been also a substantial effort to study PGG with
heterogeneous groups. However, the results are somewhat
inconsistent because of differences in implementation and various social, historical or other factors [102,110,111]. In an early
review of experimental work on PGG, Ledyard [109] concluded that asymmetry of benefits had negative effects on
contributions. Fisher et al. [112] showed that individuals with
lower valuation to cost ratios vi/ci contribute less than those
with high values of this ratio, as predicted by the theory.
Chan et al. [113] provided some experimental support for an
increase in public good provision with inequality in wealth,
again as predicted. However, in the experiment conducted
by Buckley & Croson [110], less wealthy individuals contributed the same absolute amount (or more as a percentage of their
income) as the more wealthy. In Anderson et al. [111], inequality reduced contributions to a public good for all group
members, regardless of their relative position. In Reuben &
Riedl [114], individuals differed in their valuation vi of
reward. With no punishment of free-riders, total contributions
were higher with inequality. Punishment did not significantly
increase total earnings but strongly increased inequality at the
cost of low-benefit members. Consequently, with punishment,
low-benefit members did not benefit from being part of a
privileged group. Secilmis & Güran [115] studied the effects
of differences in endowment. They observed higher average
contributions in egalitarian groups. With inequality, highendowment individuals contributed more in the absolute
sense but approximately the same percentage of their endowment; contributions by individuals with the same endowment
decreased with increasing group inequality. Diekmann and
co-workers [116,117] studied the asymmetric Volunteer’s
Dilemma with punishment. In their experiments, withingroup social dynamics led to a state where punishment was
mostly administered by individuals for whom it was less
costly. Kölle [78] allowed for differences in valuation vi and
capabilities/strengths si. Without punishment, heterogeneity
in valuations did not increase the average effort but that in capabilities did increase it. Punishment was much more effective
with variation in capabilities than with that in valuations or
with no variation. As noted above, with quadratic benefit and
cost functions, individuals are predicted to contribute in proportion to their benefit to cost ratios. Experimental results by
McGinty & Milam [102] largely support this prediction.
(b) Us versus them games
A limitation of Olson’s theory is that he considered the CAP
as a phenomenon within a single group assuming that
8
Phil. Trans. R. Soc. B 370: 20150016
(c)
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0.5
1.0
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Xj
Pj ¼ P ,
Xk
ð2:5Þ
(i) Homogeneous groups
Katz et al. [124] (see also [76,125]) studied a conflict between
two groups for a prize. They assumed that individual costs
grew linearly with the efforts and that the reward was divided
equally within the winning group. They also allowed for differences in group sizes and in how groups valued the prize.
A conclusion of their analysis is that at a NE, the group in
which individuals have a higher valuation of the prize, will
expend more effort and will have a higher probability of winning. This implies that a smaller group in which individuals
have higher valuation of the prize (because, in the case of success, each group member would get a larger amount) will have
a higher probability of winning. This is another example of
Olson’s group size paradox but in the context of betweengroup contests. With multiple groups of equal size n, each
of which values the prize equally, the equilibrium group
effort is X ¼ B/nc [76]. Individual contributions are not
defined uniquely, but if each group member contributes
equally, x ¼ X/n. However, if individual costs are certain
nonlinear functions of efforts, an increase in the group size
can raise the group effort [126].
So far, I have discussed models with endogenously specified valuations/shares. It is also possible that individual
shares are defined by outcomes of an additional within-group
conflict. Such nested or multi-level contests have also been
studied [118,123,127–130] with a general conclusion that external conflicts cause increasing within-group cooperation and
reduced free-riding.
(ii) Heterogeneous groups
Baik [131] generalized the Katz et al. [124] model for a case of a
multi-group contest in which group members differ in their
valuations (or shares) of the prize (see also [76]). His prediction
is that within each group only an individual with the highest
valuation of the prize will make a positive contribution while
the rest will contribute nothing. This result implies that neither
the group size nor the distribution of valuations among n – 1
other group members matter. However, if individual efforts
are limited from above by a certain exogenously specified
value, then multiple individuals can become contributors
within each group and alternative equilibria with different
sets of contributors become possible [34,131,132]. Epstein &
Mealem [133] studied a contest between two groups. They
showed that if individual costs are specific nonlinear functions
of individual efforts, then all group members contribute in
(iii) Examples
To illustrate and extend these results, I will use contest
success function (2.5) assuming no variation in baseline
pay-offs ( fi,0 ¼ 1), but within-group heterogeneity in benefit
and cost parameters which I will write as biG and ci. (The
reason for scaling the benefit parameter by G is that with
similar group efforts Pj 1=G, so that the expected benefit
per group is bi as in ‘us versus nature’ games.)
Basic model. Consider the basic model with linear impact and
cost functions, and no group selection [131]. In this model, the
relevant benefit-to-cost ratios are ri ¼ bi/ci; only the individual
with the highest ri value always makes an effort x1 ¼ r1 while
all other group-mates always free-ride. Figure 5a illustrates
this model numerically for the case of equal costs (ci ¼ 1).
As predicted, there is a single contributor—the highest
valuator—whose effort increases with within-group inequality.
Note a sharp decline in fertility of the highest valuator with
inequality which can be substantially smaller than that of
other individuals. That is, the highest valuator behaves as
an altruist.
Adding group extinction to the basic model by using payoff function (2.3b) leads to a model studied by Gavrilets &
Fortunato [77]. As mentioned above, in their model, there
is a threshold valuation vcrit so that only individuals with
vi . vcrit will contribute at ESS while all other group-mates
will free-ride. The group effort increases with inequality.
Fitness of high valuators can be smaller than that of low
valuators in spite of the fact that the former are getting the
biggest share of the reward. Under some conditions, fitness
of the highest valuators can be very close to zero so that
they can be viewed as effectively sacrificing themselves for
the benefit of their groups. Figure 5b illustrates this model
numerically assuming all ci ¼ 1.
The behaviour of the highest valuators may seem altruistic but, as explained in [77], actually it is not. For example,
in the case of hierarchical groups, dominant individuals
maximize their fitness by contributing; given the subordinates do not contribute at all, dominants will not be better
off by reducing their contribution. Thus, the non-contributors
are indeed free-riding, but the contributors are not altruistic; paradoxically, they are acting in their own interest by
contributing to the collective good. What is driving their
9
Phil. Trans. R. Soc. B 370: 20150016
[76,119,120], which is also used in evolutionary biology
models [34,84,107,121 –123]. Note that the denominator in
P
the equation above can be written as Xj þ k=j Xk : This
implies that functions (2.4) and (2.5) become identical if we
interpret half-success parameter X0 as the total effort of all
P
other groups in the population, j=i Xj :
proportion to their valuations and free-riding is reduced. Similar behaviour is predicted for the CES impact function
[101,134]. In the case of weak-link contests, analysis predicts
multiple equilibria including those with no free-riding at all
[135]. In best-shot contests, there can be ‘perverse equilibria’
in which the highest valuation players completely free-ride
on others by exerting no effort [132]. Allowing for group
extinction results in that multiple individuals with highest
valuations start contributing rather than a single individual
[77]. The latter paper also predicts an increased group effort
with within-group inequality and reduced net benefit (fertility)
of high valuators relative to their free-riding group-mates
(see below and also [136]). Choi et al. [137] studied a model
of a contest between two groups of size n ¼ 2 in the presence
of both within-group power asymmetry and conflict over the
share of the reward. They showed that high within-group
inequality can increase the group effort in external conflict but the effect depends on the degree of synergicity of
individual contributions.
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external threat to the group does not exist or can be treated as
constant [118]. Here I review the us versus them games that
explicitly capture the effects of other groups on the degree/
probability Pj of the focal group’s success. The most commonly used function for specifying Pj in us versus them
games in the economics literature is the Tullock contest
success function
4
(b) 4
2
0
n=4
n=8
fertility
fertility
n=4
n=8
n = 16
n=4
n=8
n = 16
1.0
1.0
0.5
0
2
0
n = 16
10
n=4
n=8
0.5
0
n = 16
(a)
(b)
2
efforts
efforts
2
1
0
n=4
n=8
0
n = 16
0.5
n=4
n=8
n = 16
n=4
n=8
n = 16
n=4
n=8
n = 16
n=4
n=8
n = 16
(d)
4
efforts
efforts
n=8
0.5
0
n = 16
(c)
2
0
n=4
n=8
2
fertility
1.0
0.5
0
4
0
n = 16
1.0
fertility
n=4
1.0
fertility
fertility
1.0
0
1
n=4
n=8
n = 16
0.5
0
Figure 6. Effects of nonlinear costs in the ‘us versus them’ game. First column (a,c): g ¼ 1.5, second column (b,d): g ¼ 2, second column g ¼ 2.5. (a,b) No
group extinction, (c,d ) with group extinction. (Online version in colour.)
contribution is that they are essentially competing with their
counterparts in other groups rather than with their own
group-mates.
Nonlinear costs. Assuming nonlinear costs term cxgij has a
qualitatively similar effect to that in us versus nature games.
P
The relevant benefit-to-cost ratios are ri ¼ bi/ci and R ¼ ri :
The group effort X solves the equation gXg ¼ Rng2 Vg1
1=ðg1Þ
with individual efforts being proportional to ri
: ffiffiffiffiffiffiffiffi
For
p
example, with g ¼ 2, at an ESS the group effort is X ¼ R=2
and each group member contributes: xi ¼ ðri =RÞX: Note that
X does not depend on the distribution of ri within the group
or the group size n. The total cost spent by the group is
cX 2 ¼ B/2, that is, one-half the overall benefit. Individual
pay-offs depend linearly on valuations. With 1 , g , 2, the
group effort decreases with group size n and increases with
variation in valuations vi. With g . 2, the situation is reversed.
Figure 6a,b illustrates these results numerically for the case
of equal costs parameters (ci ¼ 1). Allowing for group extinction (i.e. using fitness function (2.3b)) results in increasing
individual efforts (figure 6c,d). The effects of within-group
inequality greatly decrease, while those of group size diminish
for g ¼ 1.5, but are augmented for g ¼ 2.5.
Synergicity. Using impact function (2.1b) has a qualitatively
similar effect to that in us versus nature games. The predicted
group effort X is X ¼ Rna2 Va1 and individual efforts are
a=ða1Þ
proportional to ri
: For example, if a ¼ 2, the group effort
at ESS is X ¼ R and each group member contributes effort
P
xi ¼ r2i X: The total cost paid by the group is C ¼ B r2i which
is minimized in egalitarian groups (i.e. with ri ¼ 1/n). Individual
pay-offs are fi ¼ 1 þ Bri ð1 vi Þ: With 1 , a , 2, the group
effort decreases with group size n. With a . 2, the situation
is reversed.
Figure 7a,b illustrates this model numerically for the
case of equal costs parameters (ci ¼ 1). The effects of allowing
for group extinction are similar to those in other models
(figure 7c,d).
Conclusions on examples. Overall, the behaviour of these
models parallels that of us versus nature models considered
Phil. Trans. R. Soc. B 370: 20150016
Figure 5. Collective action in the ‘us versus them’ game with production function (2.5) without (a) and with (b) group extinction. Additive impact function, linear
costs function. (Online version in colour.)
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(a)
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10
2
1
0
n=4
n=8
n = 16
(b)
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4
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0.1
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n=8
0
10
(d )
4
n=4
n=8
n = 16
1
n = 16
n=4
n=8
n = 16
8
0.1
100
4
0
10
n=4
n=8
n = 16
n=4
n=8
n = 16
1
1.0
fertility
1.0
fertility
n=8
0.5
n = 16
efforts
fertility
efforts
n=4
8
0.5
0
n=4
1.0
0.5
0
1
n=4
n=8
n = 16
0.5
0
Figure 7. Effects of synergicity in the ‘us versus them’ game. First column (a,c): a ¼ 1.5, second column (b,d): a ¼ 2, second column a ¼ 2.5. (a,b) No group
extinction, (c,d ) with group extinction. The dashes show the group effort on the logarithmic scale specified on the right y-axis. (Online version in colour.)
above but individual and group efforts are always higher. In us
versus nature games, groups often cooperate only if a corresponding benefit-to-cost ratio exceeds a certain threshold. By
contrast, in us versus them games, groups always contribute a
non-zero effort. Modelling predictions depend on benefit-tocost ratios ri ¼ bi =ðci X0 Þ or ri ¼ bi =ðci X02 Þ in us versus nature
games and ri ¼ bi =ci in us versus them games. We can compare
these two types of games directly by setting X0 ¼ 1. Then in the
basic model, in games against nature collective good is propffiffiffiffi
duced if r1 . 1 and the total group effort is r1 1: In us
versus them games, groups always make an effort equal to r1.
With quadratic costs, the group p
efforts
ffiffiffiffiffiffiffiffi are a solution of the
equation 2XðX þ 1Þ2 ¼ R and
R=2, respectively, where
P
R ¼pffiffiffiffi ri : With collaborative ability a ¼ 2, the group efforts
are R 1 (which requires R . 1) and R, respectively. In all
these cases, group efforts in us versus them games are higher
than in us versus nature games. I conclude that direct competition with other groups is much more conducive for the
evolution of cooperation than collaboration against nature.
(iv) Experimental games
Experimental work on contests including between-group contests was recently reviewed comprehensively by Dechenaux
et al. [138]. A general conclusion of their analysis is that
between-group contests greatly increase individual efforts
(up to five times the Nash equilibrium prediction) and mitigate
the within-group free-rider problem. A number of experimental studies both in the economics and evolutionary biology
literatures have also incorporated group selection in their
design by either penalizing groups making the lowest efforts
or rewarding groups making the highest efforts [139–143].
These studies show that adding group selection further
increases both individual and group efforts.
Most experimental studies assume identical players
within each group. One notable exception is Sheremeta
[144], who allowed for within-group variation in valuation.
Sheremeta found that with a linear production function, all
players expend significantly higher efforts than predicted
by theory. In best-shot contests, most of the effort is expended
by strong players, whereas weak players free-ride. In weakest-link contests, there is almost no free-riding and all
players expend similar positive efforts. Experiments also
strongly confirm that individual efforts are higher when
members of the group are rewarded proportionally to their
efforts rather than equally [127,145,146].
3. Discussion
The ability of groups to be successful in potentially profitable
collective actions, aiming to overcome nature challenges or
win competition with neighbouring groups of co-specifics, is
undermined by a number of factors. One is the possibility of
free-riding. If individual efforts are costly and at least some
benefits of a collective action can be enjoyed independently
of the level of participation, individuals can withdraw their
effort to take advantage of this situation or avoid being taken
advantage of by others. Another factor is within-group heterogeneity in various characteristics that affect decisions of
individuals, such as the valuation or share of the prize, cost
of efforts, individual capability, strength and personality.
Such heterogeneity, which is ubiquitous both in animal and
human groups, augments the conflict between individual
interests. There is also a need to be able to efficiently coordinate
group activities. Moreover, the success of collective actions is
affected by exogenous and/or stochastic factors.
A substantial effort of theoreticians in several disparate
research areas has been devoted to understanding the CAP
in heterogeneous groups. The existing body of work
reviewed above allows one to make certain generalizations
about factors affecting individual efforts and group success
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that both the first and second Olson’s insights require certain
qualifications (as was already stressed earlier [8,12]).
The effects of parameters and modelling assumptions in
the two types of games considered here are qualitatively similar. However, individual and group efforts are much lower in
us versus nature games than in us versus them games. In the
former, the success of one group in a collective goods production does not affect that of another group. By contrast,
in the latter one group’s success means another group’s failure. This difference results in stronger natural selection in the
us versus them models which in turn produces a larger evolutionary response in individual contributions. In us versus
nature games, groups under-contribute, so that the group
effort is smaller than what would be an optimum for the
group. By contrast, in us versus them games, groups overcontribute as they would be better off if the efforts spent on
between-group competition were minimized. Allowing for
group extinction greatly increases individual contributions
in both types of games.
Most of my discussion focused on within-group differences
in cost–benefit ratios, shares of the reward or endowments.
However, within-group differences in strengths or capabilities
will result in similar effects. That is, stronger or more capable
group members are predicted to be the biggest contributors
[77,78]. I also assumed that groups had similar size, structure
and that they valued the reward equally. The differences in
valuation and sizes between groups are converted into the
differences in valuation between individuals from different
groups; individuals (and groups) with higher valuations will
make more effort as discussed above. Another possible extension concerns production function (2.4) and contest success
function (2.5) which can be generalized to Pj ¼ Xjb =ðXjb þ X0b Þ
P
and Pj ¼ Xjb = Xkb , respectively, where b is a ‘decisiveness’
parameter. Large values of b imply that small changes in
group efforts can cause significant changes in Pj. Increasing b
above one does not change the results qualitatively but
increases individual efforts [77,71].
Models show that with certain nonlinear cost and impact
functions, efficient within-group collaboration becomes possible. The important factor here is not nonlinearity per se, but
the underlying assumptions. For example, with quadratic costs
( g ¼ 2), small individual efforts are very ‘cheap’, while large
individual efforts are very ‘expensive’. Similarly, with large
synergicity parameter a, a substantial group effort can emerge
from small individual contributions. These are conditions
simplifying collective actions with significant participation.
Full participation can also be promoted by within-group
communication [149] and punishment [150].
Genetic relatedness is a powerful force promoting
cooperation [40,86,151]. Some theoretical studies reviewed
above have explicitly incorporated genetic relatedness in
mathematical models [68,77,85,104]. The results are in line
with the expectation that genetic relatedness simplifies collective action. However, assuming realistic population structure
and significant dispersal of offspring of one sex leads to relatively low levels of within-group genetic relatedness [77,152]
and, correspondingly, low effects on collective actions.
In us versus them games, the group effort X can be viewed
as a proxy for intensity of between-group competition.
The models thus predict that the latter would increase with
larger rewards, smaller costs and higher synergicity of individual efforts. As discussed above, the effects of group size,
within-group heterogeneity and the type of costs vary.
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in collective actions. Theoretically, individual efforts, the
amount of within-group free-riding and the likelihood of
the group’s success depend on (i) the group size and composition, (ii) how efforts of individuals are converted into fitness
costs, (iii) how efforts of individual group members are aggregated into a group effort, (iv) how the latter is translated into
the group’s success, (v) how success in a collective good production affects the group’s survival, and (vi) how members of
surviving groups divide the prize if successful. Information on
these aspects of collective actions is needed if one is to apply
theoretical predictions to specific groups, populations or species.
A very common theoretical conclusion is that the largest
beneficiaries of collective goods will make the largest contributions towards their production (but see [73,147]). Some
models, e.g. with linear impact and cost functions, predict
disproportionately large contributions of the ‘great’ who are
exploited by the ‘small’ [5]. In these models, there can be
one or few contributors per group while the rest free-ride.
Other nonlinear models predict that all group members will
contribute proportionally to their endowments or the ratio
of the valuation of the prize/goods to the cost of individual
effort. In humans, high contributors can also increase their
reputation which would allow them to get better mates and
allies [148].
A paradoxical novel prediction of recent work is that
group members who are getting the biggest share of the
reward or who value the reward the most can end up
with the smaller net pay-offs than the rest of the group
([77] and above). This happens because such individuals
make disproportionately high efforts and, as a result, pay
very high costs. The possibility of group extinctions augments these effects. Intuitively, for the case of groups
with a dominance hierarchy, once such a hierarchy is established and the shares of the reward going to dominant
individuals are fixed, they are mostly competing not against
their group-mates but against their peers in other groups. This
between-peer competition drives the evolution of increasingly
large efforts. Under some conditions, high valuators have
essentially zero fitness, that is, they effectively sacrifice themselves for the benefit of the group. This self-sacrifice is not
opposed by selection because, first, it is expressed conditionally on the rank (which is assigned randomly) and,
second, it is not subject to within-group selection (only to
between-group selection).
The group effort in a collective action (as well as the
degree/probability of its success) increases with the benefit
and decreases with the cost parameters. This is intuitive.
The effect of the group size on the group effort is complex.
Group size can have negative, zero or positive effect on the
group production. For example, members of small groups
can value the same reward more than members of large
groups (because their absolute shares will be larger). This
would result in larger efforts of smaller groups. However, if
individual costs grow rapidly with their efforts, or if there
is a hard limit on individual efforts, or if a goal of a collective
action simply cannot be accomplished by a single individual
or few individuals, larger groups will be at an advantage
(because small individual efforts can be ‘compensated’ by a
large number of contributors). Similarly, the effects of
within-group heterogeneity on the group effort vary. They
are positive in linear models, absent in some nonlinear
models and negative in models with high synergicity or
strong nonlinearity in cost functions. Overall, I conclude
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(a) Applications to mammals
Theoretical results have some implications for the origins
of successful collective actions in human ancestors. Models
show that in games against nature, cooperation becomes
successful if the corresponding benefit/cost ratio R is high
enough, and that degree of success greatly increases with collaborative ability a. From a modelling perspective, successful
large-game hunting may become possible after some technological changes, such as an invention of spears, that would
increase the ability to kill large animals (reducing half-success
effort X0) and/or decrease the danger to hunters (reducing cost
parameters ci). In competition against other groups, technological innovations allowing for better defence of a valuable
territory and/or resource (increasing B) would cause increasing individual efforts. Direct competition against other
groups also strongly promotes within-group collaboration.
Moreover, it is more likely to result in group extinction which
would augment this effect. Similar effects will follow worsening environmental conditions [158]. All these factors will
increase intensity of group selection which in turn would promote collaboration in both types of games. A similar effect will
be achieved by an increase in collaborative ability (increasing
a) with the emergence of language. (Gavrilets [71] models
the evolution of a explicitly). An additional factor could be
the appearance of certain behavioural strategies, as a result of
technological and/or cultural innovation. For example, raids
usually have low individual costs (low c) but large benefit
The work reviewed here has a number of implications for
behaviour of modern humans. First, it predicts that humans
have a genetic predisposition for collaborative group activities. This is in line with a consistent observation that human
infants are motivated to collaborate in pursuing a common
goal [161] and that cooperative acts result in activation of
brain regions involved in reward processing, independently
of material gains [162]. People cooperate when groups face
failure because of external threats, e.g. harsh environmental conditions or natural disasters [163,164]. However, as
predicted by the theory above, cooperation increases dramatically in the presence of direct between-group competition (see
experimental economics work discussed above) to a level
that ‘cues of group competition have an automatic or unconscious effect on human behaviour that can induce increased
within-group cooperation’ [165]. A number of other observations about human psychology (e.g. in-group/out-group
biases, widespread obsession with team sports, and sex differences in the motivation to form, and skill at maintaining, large
competitive groups [166]) strongly support the idea about the
importance of between-group conflicts in shaping human
social instincts.
Existing experimental data on the effects of within-group
heterogeneity are also largely in line with theoretical predictions (see above). Additionally, it has been shown that
more competitive, individualistic players contribute more to
collective goods with group competition [167] and that individuals of high status contribute more towards group
goals [168]. Models predict that individuals who find themselves in a leadership position will exhibit more propensity
for pro-social and self-sacrificing actions in warfare and
other between-group competition scenarios. In line with
these expectations in some human groups, the most aggressive warriors have lower reproductive success than other
men, as documented for the horticulturalist/forager Waorani
of Ecuador [169] and Nyangatom of Ethiopia [170]. The
Cheyenne war chiefs were expected to be killed in combat,
and leaders in the Nyangatom of Ethiopia, the Kapauka of
New Guinea and the Jie of Uganda all take greater risks in
combat [171,172]. The data show that the highest mortality
in the US Army in the Iraq war was among First and
Second Lieutenants, who typically lead combat patrols
[173]. In recent public goods experiments [117], actors tacitly
coordinated on the strongest group member to punish defectors, even if the strongest individual incurs a net loss from
punishment. These experiments showed that an arbitrary
assignment of an individual to a focal position in the social
hierarchy can trigger changes in his/her behaviour leading
to the endogenous emergence of more centralized forms of
punishment. Note that models predict higher efforts and
costs not only for individuals who actually hold or have been
assigned high rank or status, but also for ‘potential alphas’
who have higher valuation of the prize, lower costs or higher
strength. Finally, modelling work suggests that bullish and
‘unethical’ behaviour by high-rank individuals towards their
Phil. Trans. R. Soc. B 370: 20150016
(b) Applications to human origins
(c) Applications to human psychology
13
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In mammals, collective actions mostly concern territorial behaviour, hunting and cooperative breeding. CAPs have been
identified in lions, wolves, dogs and many primates. For
example, capuchines are more likely to run away from territorial intrusions when their group has a numeric advantage; each
one-individual increase in relative group size raises the odds of
flight by 25% [17]. Individuals also base their decisions on the
value of the territory, which is higher for the central parts of
the group’s range relative to the periphery. In capuchines, the
probability that a focal animal fled from a simulated intrusion
by a neighbouring group was 91% lower in experiments that
occurred in the centre compared with on the edge of its
group’s range, whereas the odds that it rushed to defend
its range were more than sixfold higher. Willems et al. [18]
argue that CAP affects territorial behaviour of approximately
30% of 135 social primates species in their sample, resulting
in a sub-optimal defence of a common range or territory.
Further analysis has led Willems & van Schaik [19] to conclude
that the only species that do not succumb to the CAP are those
that live in relatively small groups with few individuals of the
dominant sex, and are characterized by philopatry of this
dominant sex or are cooperative breeders.
A number of empirical studies support the prediction of
higher effort for high-rank individuals. In chimpanzees, highrank males travel further into the periphery during border
patrols [153] and males with higher mating success are more
likely to engage in this activity [154], which is energetically
costly [155]. In ring-tail lemurs [14] and blue monkeys [156],
high-rank females participate more in the defence of communal feeding territories than low-rank females. In meerkats,
dominant males respond more strongly to intruder scent
marks [157]. High-rank chacma baboon males are more likely
than low-rank males to join inter-group loud call displays [16].
(large B) [159,160]. Adapting raids as a group strategy would
promote more efficient collaboration. Once individuals
develop abilities to cooperate effectively in some specific
types of activities (like hunting or raiding), their skills can be
successfully applied to many other collective actions.
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(d) Final thoughts
Competing interests. I have no competing interests.
Funding. This work was supported in part by the National Institute for
Looking at the overall picture emerging from the work
reviewed here, to what extent do groups solve the CAP? The
answer naturally depends on what one means by ‘solving
the CAP’. In most cases of us versus nature games with large
enough benefit-to-cost ratio and in all cases of us versus them
games discussed above, group efforts are positive. Therefore
in a weak sense, groups almost always solve the CAP.
Mathematical and Biological Synthesis through NSF Award no. EF0830858, with additional support from The University of Tennessee,
Knoxville and by the U.S. Army Research Laboratory and the
U. S. Army Research Office under grant number W911NF-14-1-0637.
Acknowledgements. I thank M. Duwal Shrestha for help with numerical
simulations and graphs. I thank L. Glowacki, R. Sheremeta,
K. Rooker, C. von Rueden and reviewers for comments, suggestions
and discussions.
References
1.
Trivers R. 1985 Social evolution. Menlo Park, CA:
Benjamin/Cummings.
2. Jens Krause GDR. 2002 Living in groups. Oxford, UK:
Oxford University Press.
3. Majolo B, Vizioli ADB, Schino G. 2008 Costs and
benefits of group living in primates: group size
effects on behaviour and demography. Anim.
Behav. 76, 1235–1247. (doi:10.1016/j.anbehav.
2008.06.008)
4. Tucker A. 1950 A two-person dilemma. In Readings
in games and information (ed. E Rasmussen),
pp. 7 –8. Oxford, UK: Blackwell, 2011.
5. Olson M. 1965 Logic of collective action: public goods
and the theory of groups. Cambridge, MA:
Harvard University Press.
6. Hardin G. 1968 Tragedy of commons. Science 162,
1243–1248. (doi:10.1126/science.162.3859.1243)
7. Hardin R. 1982 Collective action. Baltimore, MD:
John Hopkins University Press.
8. Sandler T. 1992 Collective action: theory and
applications. Ann Arbor, MI: University of Michigan
Press.
9. Marwell G, Oliver P. 1993 The critical mass in
collective action. Cambridge, UK: Cambridge
University Press.
10. Medina LF. 2013 The analytical foundations of
collective action theory: a survey of recent
developments. Annu. Rev. Polit. Sci. 16, 259 –283.
(doi:10.1146/annurev-polisci-032311-110742)
11. Ray G. 2013 Foundations of collective action: towards
a general theory. In Social capital and rural
development in the knowledge society (eds H
Westlund, K Kobayashi), pp. 75–109. Google eBook.
12. Pecorino P. 2015 Olson’s logic of collective action at
fifty. Public Choice 162, 243 –262. (doi:10.1007/
s11127-014-0186-y)
13. Nunn C. 2000 Collective action, free-riding, and
male extra-group conflict. In Primate males
(ed. P Kappeler), pp. 192 –204. Cambridge, UK:
Cambridge University Press.
14. Nunn C, Deaner R. 2004 Patterns of participation and
free riding in territorial conflicts among ringtailed
lemurs (Lemur catta). Behav. Ecol. Sociobiol. 57,
50–61. (doi:10.1007/s00265-004-0830-5)
15. Nunn C, Lewis R. 2006 Cooperation and collective
action in animal behavior. In Economics in nature
(eds R Noë, JARAM van Hooff, P Hammerstein), pp.
42 –66. Cambridge, UK: Cambridge University Press.
16. Kitchen DM, Beehner JC. 2007 Factors affecting
individual participation in group-level aggression
among non-human primates. Behaviour 144,
1551 –1581. (doi:10.1163/156853907782512074)
17. Crofoot MC, Gilby IC. 2012 Cheating monkeys
undermine group strength in enemy territory. Proc.
Natl Acad. Sci. USA 109, 501 –505. (doi:10.1073/
pnas.1115937109)
18. Willems E, Hellriegel B, van Schaik CP. 2013 The
collective action problem in primate territory
economics. Proc. R. Soc. Lond. B 280, 20130081.
(doi:10.1098/rspb.2013.0081)
19. Willems E, van Schaik CP. 2015 Collective action end
the intensity of between-group competition in
nonhuman primates. Behav. Ecol. 26, 625 –631.
(doi:10.1093/beheco/arv001)
20. Packer C, Ruttan L. 1988 Evolution of cooperative
hunting. Am. Nat. 132, 159–198. (doi:10.1086/284844)
21. Bonanni R, Valsecchi P, Natoli E. 2010 Pattern of
individual participation and cheating in conflicts
between groups of free-ranging dogs. Anim. Behav.
79, 957 –968. (doi:10.1016/j.anbehav.2010.01.016)
22. MacNulty DR, Smith DW, Mech LD, Vucetich JA,
Packer C. 2012 Nonlinear effects of group size on
the success of wolves hunting elk. Behav. Ecol. 23,
75– 82. (doi:10.1093/beheco/arr159)
23. MacNulty DR, Tallian A, Stahler DR, Smith DW. 2014
Influence of group size on the success of wolves
hunting bison. PLoS ONE 9, e112884. (doi:10.1371/
journal.pone.0112884)
24. Mathew S, Boyd R. 2011 Punishment sustains largescale cooperation in prestate warfare. Proc. Natl
Acad. Sci. USA 108, 11 375– 11 380. (doi:10.1073/
pnas.1105604108)
25. Glowacki L, Wrangham RW. 2013 The role of
rewards in motivating participation in simple
warfare. Hum. Nat. 24, 444 –460. (doi:10.1007/
s12110-013-9178-8)
26. Mathew S, Boyd R. 2014 The cost of cowardice:
punitive sentiments towards free riders in Turkana
raids. Evol. Hum. Behav. 35, 58 –64. (doi:10.1016/j.
evolhumbehav.2013.10.001)
27. Hawkes K. 1992 Sharing and collective action. In
Evolutionary ecology and human behavior (eds EA
Smith, B Winterhalder), pp. 269–300. New York,
NY: Aldine de Gruyter.
28. Carballo DM (ed.). 2013 Cooperation and collective
action: archaeological perspective. Boulder, CO:
University Press of Colorado.
29. Carballo DM, Roscoe P, Feinman GM. 2014
Cooperation and collective action in the cultural
14
Phil. Trans. R. Soc. B 370: 20150016
However, in almost all cases, some free-riding (i.e. reduced
effort of some group members) is present. Therefore in a
strong sense, groups almost always succumb to the CAP. Do
the ‘great’ get exploited by the ‘small’? The answer is yes in
the sense that they usually make a larger effort in collective
good production than the ‘small’. However, in many cases,
the ‘great’ are well compensated for their effort as their
shares of reproduction are higher than those of the ‘small’.
However, in some situations the ‘great’ get exploited to the
extreme as the logic of evolutionary processes make them effectively sacrifice themselves for the benefit of their group-mates.
Theoretical predictions discussed here are directly applicable only under specific sets of conditions captured by the
corresponding models. It is important to check to what
extent they hold within other approaches to studying collective
actions mentioned at the beginning of this paper.
rstb.royalsocietypublishing.org
group-mates [174] can be accompanied by their higher
contributions towards between-group competition.
In a number of models reviewed above, members of
hierarchically structured groups align their efforts linearly proportional to their shares of the reward. This suggests that
compensation proportional to the effort may be a default
human expectation for group members who are unequal
in their roles and types of contribution to a group success.
Viewed this way, models thus provide a theoretical justification
for a major postulate of the classical equity theory [175,176] that
employees seek to maintain equity between their input-tooutput ratio and that of others, and that any variation in these
ratios between group members will be viewed as unfair treatment. (The theory does recognize that the ‘input’ of workers
includes many factors besides working hours, and that their
‘output’ includes many rewards besides money.)
Downloaded from http://rstb.royalsocietypublishing.org/ on June 16, 2017
31.
32.
33.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
games. Ecol. Lett. 8, 748–766. (doi:10.1111/j.14610248.2005.00773.x)
Archetti M, Scheuring I. 2012 Game theory of public
goods in one-shot social dilemmas without
assortment. J. Theor. Biol. 299, 9 –20. (doi:10.
1016/j.jtbi.2011.06.018)
Shutters ST. 2013 Towards a rigorous framework for
studying 2-player continuous games. J. Theor. Biol.
321, 40– 43. (doi:10.1016/j.jtbi.2012.12.026)
Doebeli M, Killingback T, Hauert C. 2013 A comment
on ‘Toward a rigorous framework for studying 2-player
continuous games’ by Shade T. Shutters, Journal of
Theoretical Biology 321, 40–43, 2013. J. Theor. Biol.
336, 240–241. (doi:10.1016/j.jtbi.2013.05.035)
Gavrilets S. 2015 Collective action and the
collaborative brain. J. R. Soc. Interface 12, 20141067.
(doi:10.1098/rsif.2014.1067)
Diekmann A. 1985 Volunteer’s dilemma. J. Confl.
Resolution 29, 605 –610. (doi:10.1177/
0022002785029004003)
Diekmann A. 1993 Cooperation in an asymmetric
Volunteer’s Dilemma game. Theory and
experimental evidence. Int. J. Game Theory 22,
75– 85. (doi:10.1007/BF01245571)
Archetti M. 2009 Cooperation as a volunteer’s
dilemma and the strategy of conflict in public goods
games. J. Evol. Biol. 22, 2192 –2200. (doi:10.1111/j.
1420-9101.2009.01835.x)
Archetti M, Scheuring I. 2010 Coexistence of
cooperation and defection in public goods games.
Evolution 65, 1140– 1148. (doi:10.1111/j.15585646.2010.01185.x)
Konrad K. 2009 Strategy and dynamics in contests.
Oxford, UK: Oxford University Press.
Gavrilets S, Fortunato L. 2014 Evolution of social
instincts in between-group conflict with withingroup inequality. Nat. Commun. 5, article 3526.
(doi:10.1038/ncomms4526)
Kölle F. 2015 Heterogeneity and cooperation: the
role of capability and valuation on public goods
provision. J. Econ. Behav. Organ. 109, 120 –134.
(doi:10.1016/j.jebo.2014.11.009)
Hirshleifer J. 1983 From weakest-link to best-shot:
the voluntary provision of public goods. Public
Choice 41, 371– 386. (doi:10.1007/BF00141070)
Hirshleifer J. 1991 The paradox of power. Econ. Politics 3,
177–200. (doi:10.1111/j.1468-0343.1991.tb00046.x)
Helmold RL. 1993 Osipov: the ‘Russian Lanchester’.
Eur. J. Oper. Res. 65, 278– 288. (doi:10.1016/03772217(93)90341-J)
Lepingwell JWR. 1987 The laws of combat?
Lanchester reexamined. Int. Secur. 12, 89 –134.
(doi:10.2307/2538918)
Gavrilets S, Duenez-Guzman EA, Vose MD. 2008
Dynamics of coalition formation and the egalitarian
revolution. PLoS ONE 3, e3293. (doi:10.1371/journal.
pone.0003293)
Gavrilets S. 2012 On the evolutionary origins of the
egalitarian syndrome. Proc. Natl Acad. Sci. USA 109,
14 069 –14 074. (doi:10.1073/pnas.1201718109)
Frank SA. 2010 A general model of the public goods
dilemma. J. Evol. Biol. 23, 1245–1250. (doi:10.
1111/j.1420-9101.2010.01986.x)
15
Phil. Trans. R. Soc. B 370: 20150016
34.
50. Myatt DP, Wallace C. 2008 An evolutionary analysis
of the volunteer’s dilemma. Games Econ. Behav. 62,
67 –76. (doi:10.1016/j.geb.2007.03.005)
51. Boyd R, Richerson P. 1992 Punishment allows the
evolution of cooperation (or anything else) in
sizable groups. Ethol. Sociobiol. 13, 171–195.
(doi:10.1016/0162-3095(92)90032-Y)
52. Fowler J. 2005 Altruistic punishment and the origin
of cooperation. Proc. Natl Acad. Sci. USA 102,
7047 –7049. (doi:10.1073/pnas.0500938102)
53. Janssen MA, Bushman C. 2008 Evolution of
cooperation and altruistic punishment when
retaliation is possible. J. Theor. Biol. 254, 541– 545.
(doi:10.1016/j.jtbi.2008.06.017)
54. Boyd R, Gintis H, Bowles S. 2010 Coordinated
punishment of defectors sustains cooperation and
can proliferate when rare. Science 328, 617 –620.
(doi:10.1126/science.1183665)
55. Mesterton-Gibbons M, Gavrilets S, Gravner J, Akcay
E. 2011 Models of coalition or alliance formation.
J. Theor. Biol. 274, 187–204. (doi:10.1016/j.jtbi.
2010.12.031)
56. Bissonnette A, Perry S, Barrett L, Mitani J, Flinn M,
Gavrilets S, de Waal FBM. 2015 Coalitions in theory
and reality: a review of pertinent variables and
processes. Behaviour 152, 1 –56. (doi:10.1163/
1568539X-00003241)
57. Perc M, Gomez-Gardenes J, Szolnoki A, Floria LM,
Moreno Y. 2013 Evolutionary dynamics of group
interactions on structured populations: a review.
J. R. Soc. Interface 10, 20120997. (doi:10.1098/
rsif.2012.0997)
58. Granovetter M. 1978 Threshold models of collective
behavior. Am. J. Sociol. 83, 1420–1443. (doi:10.
1086/226707)
59. Macy MW. 1991 Chains of cooperation: threshold
effects in collective action. Am. Sociol. Rev. 56,
730 –747. (doi:10.2307/2096252)
60. Oliver PE. 1993 Formal models of collective action.
Annu. Rev. Sociol. 19, 271– 300. (doi:10.1146/
annurev.so.19.080193.001415)
61. Heckathorn DD. 1993 Collective action and group
heterogeneity: voluntary provision versus selective
incentives. Am. Sociol. Rev. 58, 329–350. (doi:10.
2307/2095904)
62. Heckathorn DD. 1996 The dynamics and dilemmas
of collective action. Am. Sociol. Rev. 61, 250– 277.
(doi:10.2307/2096334)
63. Centola DM. 2013 Homophily, networks, and critical
mass. Rationality Soc. 25, 3– 40. (doi:10.1177/
1043463112473734)
64. Centola D. 2013 A simple model of stability in
critical mass dynamics. J. Stat. Phys. 151, 238– 253.
(doi:10.1007/s10955-012-0679-3)
65. Siegel DA. 2009 Social networks and collective
action. Am. J. Polit. Sci. 53, 122–138. (doi:10.1111/
j.1540-5907.2008.00361.x)
66. Morgenstern AP et al. 2013 Resource letter
MPCVW-1: modeling political conflict, violence, and
wars: a survey. Am. J. Phys. 81, 805 –814. (doi:10.
1119/1.4820892)
67. Doebeli M, Hauert C. 2005 Models of cooperation
based on Prisoner’s Dilemma and the Snowdrift
rstb.royalsocietypublishing.org
30.
evolution of complex societies. J. Archaeol. Method
Theory 21, 98 –123. (doi:10.1007/s10816-0129147-2)
Sahlins M. 1972 Stone age economics. Chicago, IL:
Aldine.
Sigmund K. 2010 The calculus of selfishness.
Princeton, NJ: Princeton University Press.
Leininger W. 2003 On evolutionarily stable behavior
in contests. Econ. Gov. 4, 177–186. (doi:10.1007/
s10101-002-0055-x)
Hehenkamp B, Leininger W, Possajennikov A. 2004
Evolutionary rent-seeking. Eur. J. Polit. Econ. 20,
1045–1057. (doi:10.1016/j.ejpoleco.2003.09.002)
Crowley P, Baik K. 2010 Variable valuations and
voluntarism under group selection: an evolutionary
public goods game. J. Theor. Biol. 265, 238–244.
(doi:10.1016/j.jtbi.2010.05.005)
Bowles S, Gintis H. 2011 A cooperative species:
human reciprocity and its evolution. Princeton, NJ:
Princeton University Press.
Darwin C. 1871 The descent of man, and selection in
relation to sex. London, UK: John Murray.
Robinson GE, Fernald RD, Clayton DF. 2008 Genes
and social behavior. Science 322, 896–900. (doi:10.
1126/science.1159277)
Sandholm WH. 2010 Population games and
evolutionary dynamics. Cambridge, MA: MIT Press.
Frank SA. 1995 Mutual policing and repression of
competition in the evolution of cooperative groups.
Nature 377, 520 –522. (doi:10.1038/377520a0)
Frank S. 1998 Foundations of social evolution.
Princeton, NJ: Princeton University Press.
Dionisio F, Gordo I. 2006 The tragedy of the
commons, the public goods dilemma, and the
meaning of rivalry and excludability in evolutionary
biology. Evol. Ecol. Res. 8, 321–332.
Frank SA. 2010 Demography and the tragedy of the
commons. J. Evol. Biol. 23, 32 –39. (doi:10.1111/j.
1420-9101.2009.01893.x)
Killingback T, Doebeli M, Hauert C. 2010 Diversity of
cooperation in the tragedy of commons. Biol. Theory
5, 3–6. (doi:10.1162/BIOT_a_00019)
Ruttan LM. 2006 Sociocultural heterogeneity and
the common. Curr. Anthropol. 47, 843–853.
(doi:10.1086/507185)
Ruttan LM. 2008 Economic heterogeneity and the
commons. World Dev. 35, 969–985. (doi:10.1016/j.
worlddev.2007.05.005)
McKelvey RD, Palfrey T. 1995 Quantal response
equilibria for normal-from games. Games Econ.
Behav. 10, 6–38. (doi:10.1006/game.1995.1023)
Anderson S, Goeree J, Holt C. 1998 A theoretical
analysis of altruism and decision error in public
goods games. J. Public Econ. 70, 297– 323. (doi:10.
1016/S0047-2727(98)00035-8)
Anderson S, Goeree J, Holt C. 2002 The logit
equilibrium: a perspective on intuitive behavioral
anomalies. South. Econ. J. 69, 21 –47. (doi:10.2307/
1061555)
Erev I, Roth A. 1998 Predicting how people play
games: reinforcement learning in experimental
games with unique, mixed strategy equilibria.
Am. Econ. Rev. 88, 848–881.
Downloaded from http://rstb.royalsocietypublishing.org/ on June 16, 2017
120. Münster J. 2009 Group contest success functions.
Econ. Theory 41, 345 –357. (doi:10.1007/s00199009-0479-4)
121. Hawkes K, Rogers AR, Charnov EL. 1995 The male’s
dilemma: increased offspring production is more
paternity to steal. Evol. Ecol. 9, 662–677. (doi:10.
1007/BF01237661)
122. Kokko H, Morrell LJ. 2005 Mate guarding, male
attractiveness, and paternity under social
monogamy. Behav. Ecol. 16, 724–731. (doi:10.
1093/beheco/ari050)
123. Reeve HK, Hölldobler B. 2007 The emergence of a
superorganism through intergroup competition.
Proc. Natl Acad. Sci. USA 104, 9736–9740. (doi:10.
1073/pnas.0703466104)
124. Katz E, Nitzan S, Rosenberg J. 1990 Rent-seeking for
pure public goods. Public Choice 65, 49 –60.
(doi:10.1007/BF00139290)
125. Cheikbossian G. 2012 Collective action problem:
within-group cooperation and between-group
competition in a rent-seeking game. Games Econ.
Behav. 74, 68 –82. (doi:10.1016/j.geb.2011.05.003)
126. Esteban J, Ray D. 2001 Collective action and the
group size paradox. Am. Polit. Sci. Rev. 95,
663–672. (doi:10.1017/S0003055401003124)
127. Gunnthorsdottira A, Rapoport A. 2006 Embedding
social dilemmas in intergroup competition reduces
free-riding. Organ. Behav. Hum. Decis. Process. 101,
184–199. (doi:10.1016/j.obhdp.2005.08.005)
128. Münster J. 2007 Simultaneous inter- and intragroup conflicts. Econ. Theory 32, 333 –352. (doi:10.
1007/s00199-007-0218-7)
129. Münster J, Staal K. 2011 War with outsiders makes
peace inside. Confl. Manage. Peace Sci. 28, 91 –110.
(doi:10.1177/0738894210396629)
130. Barker JL, Barclay P, Reeve HK. 2012 Within-group
competition reduces cooperation and payoffs in
human groups. Behav. Ecol. 23, 735–741. (doi:10.
1093/beheco/ars020)
131. Baik KH. 2008 Contests with group-specific publicgood prizes. Soc. Choice Welfare 20, 103–117.
132. Chowdhury SM, Topolyan I. 2013 The attack and
defense group contests. University of East Anglia AFE
Working Paper No. 49.
133. Epstein GS, Mealem Y. 2009 Group specific public
goods, orchestration of interest groups with free
riding. Public Choice 139, 357–369. (doi:10.1007/
s11127-009-9398-y)
134. Kolmar M, Rommeswinkel H. 2010 Group contests
with complementarities in effort. CESifo Working
Paper No. 3136.
135. Lee D. 2012 Weakest-link contests with groupspecific public good prizes. Eur. J. Polit. Econ. 28,
238–248. (doi:10.1016/j.ejpoleco.2011.11.003)
136. Nitzan S, Ueda K. 2014 Intra-group heterogeneity
in collective contests. Soc. Choice Welfare 43,
219–238. (doi:10.1007/s00355-013-0762-y)
137. Choi JP, Chowdhury SM, Kim J. 2010 Group
contest with internal conflict and power
inequality. See http://www.vwl1.ovgu.de/
vwl1_media/downloads/yrwocat/
Chowdhury_Group_Contest.pdf (accessed 4
November 2010).
16
Phil. Trans. R. Soc. B 370: 20150016
104. He J-Z, Wang R-W, Jensen CXJ, Li Y-T. 2012
Cooperation in an asymmetric volunteer’s dilemma
game with relatedness. Chin. Sci. Bull. 57,
1972 –1981. (doi:10.1007/s11434-012-5178-z)
105. He J-Z, Wang R-W, Li Y-T. 2014 Evolutionary
stability in the asymmetric Volunteer’s Dilemma.
PLoS ONE 9, e103931. (doi:10.1371/journal.pone.
0103931)
106. Waxman D, Gavrilets S. 2005 Target review:
20 questions on adaptive dynamics. J. Evol. Biol.
18, 1139–1154. (doi:10.1111/j.1420-9101.2005.
00948.x)
107. Gavrilets S. 2012 Human origins and the transition
from promiscuity to pair-bonding. Proc. Natl Acad.
Sci. USA 109, 9923 –9928. (doi:10.1073/pnas.
1200717109)
108. Chapais B. 2008 Primeval kinship: how pair-bonding
gave birth to human society. Cambridge, UK:
Harvard University Press.
109. Ledyard JO. 1995 Public goods: a survey of
experimental research. In hanbook of experimental
economics (ed. KJ Roth), pp. 111– 194. Princeton,
NJ: Princeton University Press.
110. Buckley E, Croson R. 2006 Income and wealth
heterogeneity in the voluntary provision of linear
public goods. J. Public Econ. 90, 935–955. (doi:10.
1016/j.jpubeco.2005.06.002)
111. Anderson LR, Mellor JM, Milyo J. 2008 Inequality
and public good provision: an experimental analysis.
J. Socio-Econ. 37, 1010 –1028. (doi:10.1016/j.socec.
2006.12.073)
112. Fisher J, Isaac RM, Schatzberg JW, Walker JM. 1995
Heterogeneous demand for public goods: behavior
in the voluntary contribution mechanism. Public
Choice 85, 249–266. (doi:10.1007/BF01048198)
113. Chan KS, Mestelman S, Moir R, Muller RA. 1996 The
voluntary provision of public goods under varying
income distribution. Can. J. Econ. 29, 54 –69.
(doi:10.2307/136151)
114. Reuben E, Riedl A. 2009 Public goods provision
and sanctioning in privileged groups. J. Conflict
Resolution 53, 72 –93. (doi:10.1177/00220027
08322361)
115. Secilmis IE, Güran MC. 2012 Income heterogeneity
in the voluntary provision of dynamic public goods.
Pac. Econ. Rev. 17, 693– 707. (doi:10.1111/14680106.12006)
116. Przepiorka W, Diekmann A. 2013 Individual
heterogeneity and costly punishment: a volunteer’s
dilemma. Proc. R. Soc. Lond. B 280, 20130247.
(doi:10.1098/rspb.2013.0247)
117. Diekmann A, Przepiorka W. 2015 Punitive
preferences, monetary incentives and tacit
coordination in the punishment of defectors
promote cooperation in humans. Sci. Rep. 5, 10321.
(doi:10.1038/srep10321)
118. Nious EMS, Tan G. 2005 External threat and
collective action. Econ. Inq. 43, 519 –530. (doi:10.
1093/ei/cbi035)
119. Tullock G. 1980 Efficient rent seeking. In Toward a
theory of the rent-seeking society (eds JM Buchanan,
RD Tollison, G Tullock), pp. 97 –112. College
Station, TX: Texas A&M University.
rstb.royalsocietypublishing.org
86. Okasha S. 2009 Evolution and the levels of selection.
Oxford, UK: Oxford University Press.
87. Weesie J, Franzen A. 1998 Cost sharing in a
Volunteer’s Dilemma. J. Confl. Resolution 42,
600–618. (doi:10.1177/0022002798042005004)
88. Zheng DF, Yin HP, Chan CH, Hui PM. 2007 Cooperative
behavior in a model of evolutionary snowdrift games
with N-person interactions. Electrophys. Lett. 80,
18002. (doi:10.1209/0295-5075/80/18002)
89. Pacheco JM, Santos FC, Souza MO, Skyrms B. 2009
Evolutionary dynamics of collective action in
n-person stag hunt dilemmas. Proc. R. Soc. B 276,
315–321. (doi:10.1098/rspb.2008.1126)
90. Hauert C, Holmes M, Doebeli M. 2006 Evolutionary
games and population dynamics: maintenance of
cooperation in public goods games. Proc. R. Soc. B
273, 2565 –2570. (doi:10.1098/rspb.2006.3600)
91. Pena J. 2011 Group-size diversity in public goods
games. Evolution 66, 623–636. (doi:10.1111/j.
1558-5646.2011.01504.x)
92. Zhang Y, Wu T, Chen X, Xie G, Wang L. 2013 Mixed
strategies under generalized public goods games.
J. Theor. Biol. 334, 52 –60. (doi:10.1016/j.jtbi.2013.
05.011)
93. Zhang Y, Fu F, Wu T, Xie G, Wang L. 2013 A tale of
two contributing mechanisms for nonlinear public
goods. Sci. Rep. 3, 2021. (doi:10.1038/srep02021)
94. Sefton M, Steinberg R. 1996 Reward structures
in public good experiments. J. Public Econ. 61,
263–287. (doi:10.1016/0047-2727(95)01534-5)
95. Laury S, Walker J, Williams A. 1999 The voluntary
provision of a pure public good with diminishing
marginal returns. Public Choice 99, 139 –160.
(doi:10.1023/A:1018302432659)
96. Laury S, Holt C. 2008 Voluntary provision of public
goods: experimental results with interior Nash
equilibria. Handb. Exp. Econ. Results 1, 792–801.
(doi:10.1016/S1574-0722(07)00084-4)
97. Doebeli M, Hauert C, Killingback T. 2004 The
evolutionary origins of cooperators and defectors.
Science 306, 859–862. (doi:10.1126/science.
1101456)
98. Geritz SAH, Kisdi E, Meszéna G, Metz JAJ. 1998
Evolutionary singular strategies and the adaptive
growth and branching of the evolutionary tree. Evol.
Ecol. 12, 35– 57. (doi:10.1023/A:1006554906681)
99. Bergstrom T, Blume L, Varian H. 1986 On the
private provision of public goods. J. Public Econ. 29,
25 –49. (doi:10.1016/0047-2727(86)90024-1)
100. Andreoni J. 1988 Privately provided public goods in
a large economy: the limits of altruism. J. Public
Econ. 35, 57– 73. (doi:10.1016/0047-2727(88)
90061-8)
101. Kölle F, Sliwka D, Zhou N. 2011 Inequality, inequity
aversion, and the provision of public goods. IZA
Discussion Paper 5514.
102. McGinty M, Milam G. 2013 Public goods provision
by asymmetric agents: experimental evidence. Soc.
Choice Welfare 40, 1159 –1177. (doi:10.1007/
s00355-012-0658-2)
103. Weesie J. 1993 Asymmetry and timing in the
volunteer’s dilemma. J. Confl. Resolution 37,
569–590. (doi:10.1177/0022002793037003008)
Downloaded from http://rstb.royalsocietypublishing.org/ on June 16, 2017
164. Milinski M, Sommerfeld RD, Krambeck H-J, Reed FA,
Marotzke J. 2008 The collective-risk social dilemma
and the prevention of simulated dangerous climate
change. Proc. Natl Acad. Sci. USA 105, 2291–2294.
(doi:10.1073/pnas.0709546105)
165. Burton-Chellew MN, West SA. 2012
Pseudocompetition among groups increases human
cooperation in a public-goods game. Anim. Behav.
84, 947 –952. (doi:10.1016/j.anbehav.2012.07.019)
166. Geary DC. 2010 The origin of mind. Evolution of brain,
cognition, and general intelligence. Washington, DC:
American Psychological Association.
167. Probst TM, Carnevale PJ, Triandis HC. 1999 Cultural
values in intergroup and single group social
dilemmas. Organ. Behav. Hum. Decis. Process. 77,
171–191. (doi:10.1006/obhd.1999.2822)
168. Simpson B, Willer R, Ridgeway CL. 2012 Status
hierarchies and the organization of collective action.
Sociol. Theory 30, 149–166. (doi:10.1177/
0735275112457912)
169. Beckerman S, Erickson PI, Yost J, Regalado J, Sparks
C, Iromenga M, Long K. 2009 Long Life histories,
blood revenge, and reproductive success among the
Waorani of Ecuador. Proc. Natl Acad. Sci. USA 106,
8134– 8139. (doi:10.1073/pnas.0901431106)
170. Glowacki L, Wrangham R. 2015 Warfare and
reproductive success in a tribal population. Proc.
Natl Acad. Sci. USA 112, 348–353. (doi:10.1073/
pnas.1412287112)
171. Glowacki L, von Rueden C. 2015 Leadership solves
collective action problems in small-scale societies.
Phil. Trans. R. Soc. B 370, 20150010. (doi:10.1098/
rstb.2015.0010)
172. McAuliffe K, Wrangham R, Glowacki L, Russel AF.
2015 When cooperation begets cooperation: the
role of key individuals in galvanizing support. Phil.
Trans. R. Soc. B 370, 20150012. (doi:10.1098/rstb.
2015.0012)
173. Buzzell E, Preston S. 2007 Mortality in American
troops in Iraq war. Popul. Dev. Rev. 33, 555 –566.
(doi:10.1111/j.1728-4457.2007.00185.x)
174. Piff PK, Stancato DM, Cóte S, Mendoza-Denton R,
Keltner D. 2012 Higher social class predicts
increased unethical behavior. Proc. Natl Acad. Sci.
USA 109, 4086 –4091. (doi:10.1073/pnas.
1118373109)
175. Adams JS. 1963 Towards an understanding of
inequity. J. Abnormal Soc. Inequality 67, 422 –436.
(doi:10.1037/h0040968)
176. Price ME. 2010 Free riders as a blind spot of equity
theory: an evolutionary consideration. In Managerial
ethics: managing the psychology of morality (ed. M
Schminke), pp. 235 –256. New York, NY: Routledge.
17
Phil. Trans. R. Soc. B 370: 20150016
152. Taylor PD, Irwin AJ, Day T. 2000 Inclusive fitness in
finite deme-structured and stepping-stone
populations. Selection 1–3, 153– 163.
153. Wilson ML, Kahlenberg SM, Wells M, Wrangham
RW. 2012 Ecological and social factors affect the
occurrence and outcomes of intergroup encounters
in chimpanzees. Anim. Behav. 83, 277– 291.
(doi:10.1016/j.anbehav.2011.11.004)
154. Watts DP, Mitani JC. 2001 Boundary patrols and
intergroup encounters in wild chimpanzees.
Behaviour 138, 299 –327. (doi:10.1163/15685
390152032488)
155. Amsler SJ. 2010 Energetic costs of territorial
boundary patrols by wild chimpanzee.
Am. J. Primatol. 72, 93 –103. (doi:10.1002/ajp.
20757)
156. Cords M. 2007 Variable participation in the defense
of communal feeding territories by blue monkeys in
the Kakamega Forest, Kenya. Behaviour 144,
1537 –1550. (doi:10.1163/156853907782512100)
157. Mares R, Young AJ, Levesque DL, Harrison N,
Clutton-Brock TH. 2011 Responses to intruder scents
in the cooperatively breeding meerkat: sex and
social status differences and temporal variation.
Behav. Ecol. 22, 594–600. (doi:10.1093/
beheco/arr021)
158. Richerson PJ, Bettinger RL, Boyd R. 2005 Evolution
on a restless planet: were environmental variability
and environmental change major drivers of human
evolution? In Handbook of evolution, vol. 2. The
evolution of living systems (eds FM Woketits, FJ
Ayala), pp. 223–242. New York, NY: Wiley.
159. Wrangham R, Jones J, Laden G, Pilbeam D, ConklinBrittain N. 1999 The raw and the stolen. Cooking
and the ecology of human origins. Curr. Anthropol.
40, 567– 594. (doi:10.1086/300083)
160. Wrangham RW, Glowacki L. 2012 Intergroup
aggression in chimpanzees and war in nomadic
hunter-gatherers evaluating the chimpanzee model.
Hum. Nat. 23, 5 –29. (doi:10.1007/s12110-0129132-1)
161. Moll H, Tomasello M. 2007 Cooperation and human
cognition: the Vygotskian intelligence hypothesis.
Phil. Trans. R. Soc. B 362, 639 –648. (doi:10.1098/
rstb.2006.2000)
162. Rilling JK. 2011 The neurobiology of cooperation
and altruism. In Developments in Primatology:
Progress and Prospects, vol. 36. Origins of altruism
and cooperation (eds RW Sussman, CR Cloninger),
pp. 295–306. Berlin, Germany: Springer.
163. Vasi I, Macy M. 2003 The mobilizer’s dilemma:
crisis, empowerment, and collective action. Soc.
Forces 81, 979–998. (doi:10.1353/sof.2003.0047)
rstb.royalsocietypublishing.org
138. Dechenaux E, Kovenock D, Sheremeta RM. In press.
A survey of experimental research on contests, allpay auctions and tournaments. Exp. Econ. (doi:10.
1007/s10683-014-9421-0)
139. Puurtinen M, Mappes T. 2009 Between-group
competition and human cooperation. Proc. R. Soc. B
276, 355–360. (doi:10.1098/rspb.2008.1060)
140. Burton-Chellew MN, Ross-Gillespie A, West SA. 2010
Cooperation in humans: competition between
groups and proximate emotions. Evol. Hum. Behav.
31, 104–108. (doi:10.1016/j.evolhumbehav.2009.
07.005)
141. Reuben E, Tyran J-R. 2010 Everyone is a winner:
promoting cooperation through all-can-win
intergroup competition. Eur. J. Polit. Econ. 26,
25 –35. (doi:10.1016/j.ejpoleco.2009.10.002)
142. Egas M, Kats R, van der Sar X, Reuben E, Sabelis
MW. 2013 Human cooperation by lethal group
competition. Sci. Rep. 3, 1373. (doi:10.1038/
srep01373)
143. Markussen T, Reuben E, Tyran J-R. 2014
Competition, cooperation and collective
choice. Econ. J. 124, F163– F195. (doi:10.1111/
ecoj.12096)
144. Sheremeta RM. 2011 Perfect-substitutes, best-shot,
and weakest-link contests between groups. Korean
Econ. Rev. 27, 5 –32.
145. Sutter M. 2006 Endogenous versus exogenous
allocation of prizes in teams: theory and
experimental evidence. Labour Econ. 13, 519–549.
(doi:10.1016/j.labeco.2005.03.001)
146. Kugler TAR, Pazy A. 2010 Public good provision in
inter-group conflicts: effects of asymmetry and
profit-sharing rule. J. Behav. Decis. Making 23,
421–438.
147. Chowdhury S, Sheremeta R, Lee D. 2013 Top guns
may not fire: best-shot group contests with groupspecific public good prizes. J. Econ. Behav. Organ.
92, 94 –103. (doi:10.1016/j.jebo.2013.04.012)
148. von Rueden C, Gurven M, Kaplan H, Stieglitz J. 2012
Leadership in an egalitarian society. Hum. Nat. 25,
538–566. (doi:10.1007/s12110-014-9213-4)
149. Cason T, Sheremeta R, Zhang J. 2012
Communication and efficiency in competitive
coordination games. Games Econ. Behav. 76,
26 –43. (doi:10.1016/j.geb.2012.05.001)
150. Abbink K, Brandts J, Herrmann B, Orzen H. 2010
Intergroup conflict and intra-group punishment in
an experimental contest game. Am. Econ. Rev. 100,
420–447. (doi:10.1257/aer.100.1.420)
151. Hamilton WD. 1964 The genetical evolution of
social behaviour I. J. Theor. Biol. 7, 1–16. (doi:10.
1016/0022-5193(64)90038-4)